Four days on from being labelled United's fourth-choice at left-back and having his professionalism questioned by Mourinho, Shaw's was introduced as a second-half substitute with Everton 1-0 up thanks to Phil Jagielka's improvised 22nd-minute finish.
And the England left-back drilled goalwards in stoppage time, with Ashley Williams sent off for handling the shot on the goalline before Zlatan Ibrahimovic converted from the penalty spot.
Ibrahimovic was back in the side following a three-game domestic suspension and had a header incorrectly ruled out for offside in the 71st minute, when it appeared United would follow Saturday's 0-0 draw against West Brom with another frustrating home stalemate.
Williams made a timely early challenge to thwart Ibrahimovic before Jesse Lingard – preferred to Henrikh Mkhitaryan and Anthony Martial in a wide attacking role by Mourinho – sliced wide from 20 yards in the 13th minute.
Ronald Koeman added some of the experience his defeated Merseyside derby line-up lacked in the form of Kevin Mirallas and Gareth Barry and the former England midfielder's lofted pass was chested down by the Belgium winger in the 21st minute, who drew a save from David De Gea at his near post.
Everton led from the resulting corner as Jagielka outmuscled Marcos Rojo to prod Williams' looping header through De Gea's legs.
United were almost level after half an hour, when Joel Robles acrobatically pushed out Daley Blind's curling free-kick and Ander Herrera clattered the rebound against the crossbar from a tight angle.
A rasping drive from Herrera forced Robles to tip behind in the 39th minute and Jagielka concluded a fine individual half in stoppage time with an immaculate challenge to halt Marcus Rashford after Ibrahimovic's flick released him into the area.
Mourinho sent on Pogba in place of Blind at the interval and the world's most expensive player was quickly into the thick of the action.
After Barry was booked for cynically barging over Rashford in full flight, Ashley Young – now operating at left-back after the half-time reshuffle – whipped in a free-kick that flicked off Romelu Lukaku and Pogba guided a header against the crossbar.
Young was booked for a rash foul on Miralles and his robust approach appeared to result in an injury, meaning Shaw entered the fray to cheers from the Old Trafford faithful.
Ibrahmovic thought he had United back on terms with 19 minutes to play as Robles failed to keep his header out but was spared as a marginal offside call fell in the visitors' favour.
Robles did not convince entirely under increasingly direct United attacks, and Fellaini fired over under pressure from Barry when the goalkeeper's punch failed to clear 73rd-minute danger.
Ibrahimovic headed beyond the crossbar as the clock ticked down but he made amends in stoppage time, sending Robles the wrong way after Shaw forced Williams into a lonely trudge from the field.
Key Opta Facts:
- Manchester United's Premier League unbeaten run now stands at 20 games, but they've drawn half of these (W10 D10).
- Phil Jagielka has scored his first Premier League goal since May 2015, 703 days ago (v Aston Villa). Jagielka is the 17th scorer for Everton this season in the Premier League – they’ve had more different scorers than any other club.
- The Red Devils have drawn 12 league games this season, their most in a campaign since the 1998/99 season (13).
- United have won just six of their 16 league games at home this season (37.5 win percentage), their worst home win percentage in a campaign since 1973/74 (33.3 per cent)
- Jose Mourinho has collected 54 points in his first 29 Premier League games as Manchester United manager, two fewer than Louis van Gaal (56).